Inventing Temperature
Page 33
Now, because all of the parameters in this equation except H are functions of air-thermometer temperature t, we can rewrite the integral in terms of t (taking H out as a constant), as follows:
where S and T
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are the temperatures of the working substance in the first and the third strokes. According to Thomson's first definition of absolute temperature, the difference between those two temperatures on the absolute scale is proportional to W/H, which can be evaluated by performing the integration in equation (11) after putting in the relevant empirical data. Comparing the absolute temperature difference estimated that way with the temperature difference (S − T) as measured on the air-thermometer scale gives the conversion factor expressing how many air-thermometer degrees correspond to one degree of absolute temperature interval, at that point in the scale.
Therefore the measurement of absolute temperature by means of the steam-water cycle came down to the measurement of the parameters occurring in the integral in (11), namely the pressure, density, and latent heat of saturated steam as functions of air-thermometer temperature. Fortunately, detailed measurements of those quantities had been made, by none other than Regnault. Using Regnault's data, Thomson constructed a table with "a comparison of the proposed scale with that of the air-thermometer, between the limits 0° and 230° of the latter." Table 4.1 gives some of Thomson's results, converted into a more convenient form. Note that the relationship between the air temperature and the absolute temperature is not linear; the size of one absolute temperature degree becomes smaller and smaller in comparison to the size of one air-temperature degree as temperature goes up. (As noted before, this absolute scale in fact had no zero point but stretched to negative infinity; that was no longer the case with Thomson's second definition of absolute temperature.)
Let us now consider whether Thomson at this stage really succeeded in his self-imposed task of measuring absolute temperature. There are three major difficulties. The first one was clearly noted by Thomson himself: the formulae given above require the values of k, the latent heat of steam by volume, but Regnault had only measured the latent heat of steam by weight. Lacking the facility to make the required measurements himself, Thomson converted Regnault's data into what he needed by assuming that steam obeyed the laws of Boyle and Gay-Lussac. He knew that this was at best an approximation, but thought there was reason to believe that it was a sufficiently good approximation for his purposes.39
Second, in calculating the amount of mechanical work, the entire analysis was premised on the assumption that the pressure of saturated steam depended only on the temperature. As noted earlier, that pressure-temperature relation was not something deducible a priori, but an empirically obtained generalization. The rigorous reliability of this empirical law was not beyond doubt. Besides, does not the
39. See Thomson [1848] 1882, 104-105. Thomson also had to use Boyle's and Gay-Lussac's laws to reason out the air-engine case; see Thomson [1849] 1882, 129, 131.
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Table 4.1. Thomson's comparison of air-thermometer temperature and his first absolute temperature Air-thermometer temperature
Absolute temperature (first definition)
0°C
0a
5
5.660
10
11.243
15
16.751
20
22.184
25
27.545
30
32.834
35
38.053
40
43.201
45
48.280
50
53.291
55
58.234
60
63.112
65
67.925
70
72.676
75
77.367
80
82.000
85
86.579
90
91.104
95
95.577
100
100
150
141.875
200
180.442
231
203.125
Source: This table, announced in Thomson [1848] 1882, 105, was published in Thomson [1849] 1882, 139 and 141, in slightly different forms than originally envisaged.
a This scale had no "absolute zero," and it was calibrated so that it would agree with the centigrade scale at 0° and 100°.
Source: Adapted from the second series of data given in Regnault 1847, 188.
use of the pressure-temperature relation of steam amount to a reliance on an empirical property of a particular substance, just the thing Thomson wanted to avoid in his definition of temperature? In Thomson's defense, however, we could argue that the strict correlation between pressure and temperature was probably presumed to hold for all liquid-vapor systems, not just for the water-steam system. We should also keep in mind that his use of the pressure-temperature relation was not in the definition of absolute temperature, but only in its operationalization. Since Carnot's theory gave the assurance that all ideal engines operating at the same temperatures had the same efficiency, calculating the efficiency in any particular system was sufficient to provide a general answer.
Finally, in the theoretical definition itself, absolute temperature is expressed in terms of heat and mechanical work. I have quoted Thomson earlier as taking comfort in that "we have, independently, a definite system for the measurement of quantities
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of heat," but it is not clear what he had in mind there. The standard laboratory method for measuring quantities of heat was through calorimetry based on the measurement of temperature changes induced in a standard substance (e.g. water), but of course that had to rely on a thermometer. Recall that Thomson's scheme for operationalizing absolute temperature was to express W/H as a function of air-thermometer temperature. A good deal of complexity would have arisen if the measure of H itself depended on the use of the air thermometer (that is, if it had to be kept inside the integral in equation (11)). In one place Thomson ([1848] 1882, 106) mentions using the melting of ice for the purpose of calorimetry, but there were significant difficulties in any actual use of the ice calorimeter (see "Ganging Up on Wedgwood" in chapter 3). Still, we could say that in principle heat could be measured by ice calorimetry (or any other method using latent heat), in which case the measure of heat would be reduced to the measure of weight and the latent heat of the particular change-of-state involved. But the last step would end up involving an empirical property of a particular substance, again contrary to Thomson's original intention!
Using Gas Thermometers to Approximate Absolute Temperature
How Thomson himself might have proposed to deal with the difficulties mentioned in the last section is an interesting question. However, it is also a hypothetical question, because Thomson revised the definition of absolute temperature before he had a chance to consider carefully the problems of operationalizing the original concept. Let us therefore proceed to a consideration of how he attempted to measure his second absolute temperature, which is expressed in equations (4) and (5). Thomson, now in full collaboration with Joule, faced the same basic challenge as before: a credible Carnot engine (or, more to the point, a reversible cycle of operations) could not be constructed in reality. The operationalization of Thomson's second absolute temperature was a long and gradual process, in which a variety of analytical and material methods were tried out by Joule and Thomson, and by later physicists. All of those methods were based on the assumption, explicit or implicit, that an ideal gas thermometer would give the absolute temperature exactly. If so, any possible measure of how much actual gases deviate from the ideal might also give us an indication of how much the temperatures indicated by actual gas thermometers deviate from the absolute temperatures.
The first thing we need to get clear about is why an ideal gas thermometer would indicate
Thomson's absolute temperature. The contention we want to support is that an ideal gas expands uniformly with absolute temperature, under fixed pressure (or, its pressure increases uniformly with temperature when the volume is fixed). Now, if we attempt to verify that contention by direct experimental test, we will get into a circularity: how can we tell whether a given sample of gas is ideal or not, unless we already know how to measure absolute temperature so that we can monitor the gas's behavior as a function of absolute temperature? Any successful argument that an ideal gas indicates absolute temperature has to be made in the realm of theory, rather than practical measurement. It is not clear to me whether
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Thomson himself made any such argument directly, but at least a clear enough reconstruction can be made on the basis of the discussions he did give.40
The argument is based on the consideration of the isothermal expansion of an ideal gas. In that process the gas absorbs an amount of heat H, while its temperature remains the same.41 The added heat causes the gas to expand, from initial volume v 0to final volume v 1 , while its pressure decreases from p 0to p 1 ; in this expansion the gas also does some mechanical work because it pushes against an external pressure. The amount of the mechanical work performed, by definition of work, is expressed by the integral ∫pdv. If we indicate Amontons temperature by t aas before, then a constant-pressure gas thermometer dictates (by operational definition) that t ais proportional to volume; likewise, a constant-volume gas thermometer dictates that t ais proportional to pressure. With an ideal gas the constant-volume and the constant-pressure instruments would be comparable, so we can summarize the two relations of proportionality in the following formula:
where c is a constant specific to the given sample of gas. That is just an expression of the commonly recognized gas law. Putting that into the expression for mechanical work, and writing W ito indicate the work performed in the isothermal process, we have:
Since we are concerned with an isothermal process, the ct aterm can be treated as a constant. So the integration gives:
If we ask how W ivaries with respect to t a , we get:
Now, the variation of W iwith temperature also has a simple relation to Carnot's function (and therefore to Thomson's second absolute temperature). Equation (2), W = QμdT, expresses the net work done in a Carnot cycle in which the temperatures of the first and third strokes (the isothermal processes) differ by an infinitesimal amount, dT. The net work in that infinitesimal cycle is (to the first
40. I am helped here by the exposition in Gray 1908, esp. 125. Gray was Thomson's successor in the Glasgow chair, and his work provides the clearest available account of Thomson's scientific work. The argument reconstructed here is not so far from the modern ones, for example, the one given in Zemansky and Dittman 1981, 175-177.
41. The same by what standard of temperature? That is actually not very clear. But it is probably an innocuous enough assumption that a phenomenon occurring at a constant temperature by any good thermometer will also be occurring at a constant absolute temperature.
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order) the infinitesimal difference between the work produced in the first stroke and the work consumed in the third stroke. So we may write:
From (2) and (16), we have:
In getting the second equality in (17) I have invoked Thomson's second definition of absolute temperature expressed in equation (4).
Now compare (15) and (17). The two equations, one in t aand the other in T, would have exactly the same form, if it were the case that JQ = W i . In other words, the Amontons temperature given by an ideal gas would behave like absolute temperature, if the condition JQ = W iwere satisfied in an isothermal expansion. But the satisfaction of that condition, according to thermodynamic theory, is the mark of an ideal gas: all of the heat absorbed in an isothermal expansion gets converted into mechanical energy, with nothing going into changes in the internal energy (and similarly, in adiabatic heating by compression, all the work spent on the gas being converted into heat). In fact, this condition is none other than what is often called "Mayer's hypothesis," because the German physician Julius Robert Mayer (1814-1878) was the first one to articulate it. Joule and Clausius also assumed that Mayer's hypothesis was empirically true, and it was the basis on which Joule produced his conjecture discussed in "Thomson's Second Absolute Temperature."42
Pondering Mayer's hypothesis led Thomson into the Joule-Thomson experiment. Thomson, who was more cautious than Joule or Clausius about accepting the truth of Mayer's hypothesis, insisted that it needed to be tested by experiment and persuaded Joule to collaborate with him on such an experiment. In an actual gas Mayer's hypothesis is probably not strictly true, which means that Amontons temperature defined by an actual gas is not going to be exactly equal to Thomson's absolute temperature. Then, the extent and manner in which an actual gas deviated from Mayer's hypothesis could be used to indicate the extent and manner in which actual-gas Amontons temperature deviated from absolute temperature.
In testing the empirical truth of Mayer's hypothesis, Joule and Thomson investigated the passage of a gas through a narrow opening, which is a process that ought to be isothermal for an ideal gas but not for an actual gas. They reasoned that an ideal gas would not change its temperature when it expands freely. But an actual gas has some cohesion, so it would probably require some energy to expand it (as it takes work to stretch a spring). If that is the case, a free expansion of an actual gas would cause it to cool down because the energy necessary for the expansion would have to be taken from the thermal energy possessed by the gas itself, unless there is
42. For a history of Mayer's hypothesis and its various formulations, see Hutchison 1976a. For an account of how Joule arrived at his conjecture, which the correspondence between Joule and Thomson reveals, see Chang and Yi (forthcoming).
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Figure 4.9. A schematic representation of the Joule-Thomson experiment (Preston 1904, 801, fig. 231).
an appropriate external source of energy. The amount of any observed cooling would give a measure of how much the actual gas deviates from the ideal.
The basic scheme for Joule and Thomson's experiment designed in order to get at this effect, often called the "porous plug experiment," was laid out by Thomson in 1851, inspired by an earlier article by Joule (1845). The procedure, shown schematically in figure 4.9, consisted in forcing a continuous stream of gas through two (spiral) pipes connected to each other through a very small orifice. Because it was difficult to measure the temperature of the gas exiting from the orifice precisely, Joule and Thomson instead measured the amount of heat that was required in bringing the gas back to its original temperature after its passage. From this amount of heat and the specific heat of the gas, the temperature at which the gas had exited from the orifice was inferred.
How were the results of the Joule-Thomson experiment used to get a measure of how much the actual gas-thermometer temperatures deviated from absolute temperature? Unfortunately the details of the reasoning are much too complicated (and murky) for me to summarize effectively here.43 What is more important for my present purpose, in any case, is to analyze the character of Joule and Thomson's results. They used the experimental data on cooling in order to derive a formula for the behavior of actual gases as a function of absolute temperature, showing how they deviated from the ideal gas law. Joule and Thomson's "complete solution" was the following:44
This equation expresses T, "the temperature according to the absolute thermodynamic system of thermometry," in terms of other parameters, all of which are presumably measurable: v is the volume of a given body of gas; p is its pressure; C is a parameter "independent of both pressure and temperature"; A is a constant that is characteristic of each type of gas; J is the mechanical equivalent of heat; and K is specific heat (per unit mass) under constant pressure. So equation (18) in principle indicates a straightforward way of measuring absolute temperature T. The second term on the right-hand side gives the measure o
f deviation from the ideal; without
43. Interested readers can review the Joule-Thomson reasoning in detail in Hutchison 1976a and Chang and Yi (forthcoming).
44. See Joule and Thomson [1862] 1882, 427-431; the equation reproduced here is (a) from p. 430.
end p.195
that term, the equation would simply reduce to the ideal gas law, which would mean that the gas thermometer correctly indicated the "temperature according to the absolute thermodynamic system of thermometry."
This marks a closing point of the problem of measuring absolute temperature. Thomson and Joule were confident that they had finally succeeded in reducing absolute temperature to measurable quantities, and they in fact proceeded to compute some numerical values for the deviation of the air thermometer from the absolute scale. The results were quite reassuring for the air thermometer (see table 4.2): although the discrepancy increased steadily as the temperature increased, it was estimated to be only about 0.4°C even at around 300°C for Regnault's standard air thermometer. One can imagine Joule and Thomson's pleasure at obtaining such a close agreement. Although their achievement received further refinement toward the end of the nineteenth century, it was never seriously challenged, to the best of my knowledge. However, it is difficult to regard Thomson and Joule's work on the measurement of absolute temperature as complete, in terms of epistemic justification. Table 4.2. Joule and Thomson's comparison of absolute temperature (second definition) and air-thermometer temperature Absolute temperature, minus 273.7°a